1. Field of the Invention
Disclosed aspects of the embodiments relate to a wavefront measuring apparatus, a wavefront measuring method, and a computer-readable medium storing a program for use in them. More particularly, the disclosed aspects relate to a wavefront measuring apparatus, a wavefront measuring method, and a program, which are adapted for deriving wavefront information from a periodic pattern (periodic pattern of light) that has been modulated by a specimen (i.e., an object to be analyzed).
2. Description of the Related Art
In the optical metrology, information of a specimen is often derived by analyzing modulation of a periodic pattern. As analytical techniques for the periodic pattern, there are known a phase shifting method and a windowed Fourier transform method. The phase shifting method and the windowed Fourier transform method are described in brief by taking, as an example, a periodic pattern I(x, y) expressed by the following formula (1). For simplicity, a one-dimensional periodic pattern is discussed below, but the following discussion is similarly applied to a two-dimensional periodic pattern as well:I(x,y)=a(x,y)+b(x,y)cos [ωxx+φ(x,y)]  (1)where a(x, y) is a background irradiance, b(x, y) is an amplitude distribution of the periodic pattern, ωx is a spatial carrier frequency, and φ(x, y) is a phase distribution or a differential phase distribution induced by the specimen.
The phase shifting method is a technique of measuring the wavefront of light having transmitted through a specimen by using a plurality of periodic patterns that have phases shifted from one another, and obtaining information of the specimen. The phase shifting method is featured in that the information of the specimen is independently obtained from the intensity of the light detected for each of pixels. Let consider, as an example, the case of extracting the information of the specimen from three periodic patterns having phases shifted from one another, which are expressed by the following formulae (2):
                                                        I              1                        ⁡                          (                              x                ,                y                            )                                =                                    a              ⁡                              (                                  x                  ,                  y                                )                                      +                                          b                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁢                              cos                ⁡                                  [                                                                                    ω                        x                                            ⁢                      x                                        +                                          φ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                                              ]                                                                    ⁢                                  ⁢                                            I              2                        ⁡                          (                              x                ,                y                            )                                =                                    a              ⁡                              (                                  x                  ,                  y                                )                                      +                                          b                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁢                              cos                ⁡                                  [                                                                                    ω                        x                                            ⁢                      x                                        +                                          φ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                              +                                          2                      ⁢                                                                                          ⁢                      π                      ⁢                                              1                        3                                                                              ]                                                                    ⁢                                  ⁢                                            I              3                        ⁡                          (                              x                ,                y                            )                                =                                    a              ⁡                              (                                  x                  ,                  y                                )                                      +                                          b                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁢                              cos                ⁡                                  [                                                                                    ω                        x                                            ⁢                      x                                        +                                          φ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                              +                                          2                      ⁢                                                                                          ⁢                      π                      ⁢                                              1                        3                                                                              ]                                                                                        (        2        )            
According to The Japan Society of Applied Physics, Atsushi Momose, Wataru Yashiro, Yoshihiro Takeda, Yoshio Suzuki and Tadashi Hattori, “Phase Tomography by X-ray Talbot Interferometry for Biological Imaging”, Japanese Journal of Applied Physics, Vol. 45, No. 6A, pp. 5254-5256 (2006), Japan, the phase distribution or the differential phase distribution induced by a specimen is obtained from the following formula (3). In the formula (3), arg[•] denotes a phase component of the expression within [ ].
                              φ          ⁡                      (                          x              ,              y                        )                          =                  arg          ⁡                      [                                          ∑                                  k                  =                  1                                3                            ⁢                                                                    I                    k                                    ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                  exp                  ⁡                                      [                                                                  -                        2                                            ⁢                                                                                          ⁢                      πⅈ                      ⁢                                                                        k                          -                          1                                                3                                                              ]                                                                        ]                                              (        3        )            
Because the obtained phase distribution or differential phase distribution is given as a value wrapped between −π and π, it is to be unwrapped.
Thus, when the periodic pattern capable of being expressed by the formula (1) is used, the wavefront information of the light having transmitted through the specimen is derived by the phase shifting method if there are three or more periodic patterns, and the phase distribution or the differential phase distribution of the specimen is obtained.
On the other hand, the windowed Fourier transform method is featured in that the wavefront information of the light having transmitted through the specimen may be derived from even one periodic pattern. Details of the windowed Fourier transform method is discussed in Elsevier, Qian Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations”, Volume 45, Issue 2, pages 304-317, Optics and Lasers in Engineering, 2007, The Netherlands. The windowed Fourier transform method is described in brief below. The formula (1) is rewritten into the following formulae (4).
                                          I            ⁡                          (                              x                ,                y                            )                                =                                    a              ⁡                              (                                  x                  ,                  y                                )                                      +                                          c                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁢                              exp                ⁡                                  [                                      ⅈ                    ⁢                                                                                  ⁢                                          ω                      x                                        ⁢                    x                                    ]                                                      +                                                            c                  *                                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁢                              exp                ⁡                                  [                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                                          ω                      x                                        ⁢                    x                                    ]                                                                    ⁢                                  ⁢                                            c              ⁢                              (                                  x                  ,                  y                                )                                      =                                                            b                  ⁡                                      (                                          x                      ,                      y                                        )                                                  2                            ⁢                              exp                ⁡                                  [                                      ⅈ                    ⁢                                                                                  ⁢                                          φ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                                              ]                                                              ,                                                    c                *                            ⁡                              (                                  x                  ,                  y                                )                                      =                                                            b                  ⁡                                      (                                          x                      ,                      y                                        )                                                  2                            ⁢                              exp                ⁡                                  [                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                                          φ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                                              ]                                                                                        (        4        )            
According to the windowed Fourier transform method, the wavefront information of the light having transmitted through the specimen is derived by locally cutting out the periodic pattern with a window function, and determining Fourier coefficients of a zeroth-order spectrum and a spatial carrier frequency. In other words, a(x, y), c(x, y) and c*(x, y) are obtained by the following formulae (5):a(x,y)=∫∫I(μ,υ)g(μ−x,υ−y)dμdυc(x,y)=∫∫I(μ,υ)g(μ−x,υ−y)exp[−iωxμ]dμdυc*(x,y)=∫∫I(μ,υ)g(μ−x,υ−y)exp[iωxμ]dμdυ  (5)where the window function is g(x, y), and μ and ν are each a variable of integration. Further, as in the phase shifting method, because the obtained wavefront information is given as a value wrapped between −π and π, it is to be unwrapped.
In the phase shifting method, when the periodic pattern is expressed by the formula (1), at least three periodic patterns having phases shifted from one another are employed for the reason that there are three unknowns. More periodic patterns are to be employed in order to obtain phase distributions or differential phase distributions in two directions from two-dimensional periodic patterns.
Meanwhile, with the windowed Fourier transform method, the information of the specimen is obtained from one periodic pattern. However, the wavefront information of the light is not independently derived from the detection result for each pixel, but by using the detection results of surrounding pixels as well. Accordingly, accuracy of the wavefront information obtained with the windowed Fourier transform method is lower than that obtained with the phase shifting method.